Probabilistic function-on-function nonlinear autoregressive model for emulation and reliability analysis of dynamical systems

2026-02-02

AI Summary

Efficient and accurate surrogate modeling of high-dimensional nonlinear dynamical systems remains a significant challenge. This work proposes the F2NARX model, which uniquely integrates function-to-function regression with probabilistic forecasting. By leveraging principal component analysis for dimensionality reduction, Gaussian process regression for surrogate modeling, and the unscented transform for autoregressive probabilistic inference, the method achieves substantial improvements in both computational efficiency—outperforming conventional NARX models by several orders of magnitude—and prediction accuracy. Notably, F2NARX requires only limited time-series data to enable active learning and deliver reliable estimates of first-passage failure probabilities.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Constructing accurate and computationally efficient surrogate models (or emulators) for predicting dynamical system responses is critical in many engineering domains, yet remains challenging due to the strongly nonlinear and high-dimensional mapping from external excitations and system parameters to system responses. This work introduces a novel Function-on-Function Nonlinear AutoRegressive model with eXogenous inputs (F2NARX), which reformulates the conventional NARX model from a function-on-function regression perspective, inspired by the recently proposed $\mathcal{F}$-NARX method. The proposed framework substantially improves predictive efficiency while maintaining high accuracy. By combining principal component analysis with Gaussian process regression, F2NARX further enables probabilistic predictions of dynamical responses via the unscented transform in an autoregressive manner. The effectiveness of the method is demonstrated through case studies of varying complexity. Results show that F2NARX outperforms state-of-the-art NARX model by orders of magnitude in efficiency while achieving higher accuracy in general. Moreover, its probabilistic prediction capabilities facilitate active learning, enabling accurate estimation of first-passage failure probabilities of dynamical systems using only a small number of training time histories.

Problem

• surrogate modeling
• dynamical systems
• nonlinear autoregressive
• function-on-function regression
• reliability analysis

Innovation

• Function-on-Function Regression
• F2NARX
• Probabilistic Emulation
• Unscented Transform
• Gaussian Process Regression